Alternative methods to estimate the State Price Density

In this paper, we propose different approaches to estimate the state price density under the classical hypothesis of the Black and Scholes model. In particular, we use the nonparametric approaches based on kernel and on the conditional expectation estimators. Firstly, we examine the real mean return function using local polynomial smoothing technique. Then, we evaluate the conditional expectation using the real probability density. According the hypothesis of the Black and Scholes model, we are able to derive a closed formula for approximating the conditional expectation with the risk neutral probability. Finally, we propose a comparison among two estimators of the state price density.